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Theorem sotri2 5963
Description: A transitivity relation. (Read 𝐴𝐵 and 𝐵 < 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
soi.1 𝑅 Or 𝑆
soi.2 𝑅 ⊆ (𝑆 × 𝑆)
Assertion
Ref Expression
sotri2 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Proof of Theorem sotri2
StepHypRef Expression
1 soi.2 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
21brel 5588 . . . 4 (𝐵𝑅𝐶 → (𝐵𝑆𝐶𝑆))
32simpld 498 . . 3 (𝐵𝑅𝐶𝐵𝑆)
4 soi.1 . . . . . . 7 𝑅 Or 𝑆
5 sotric 5470 . . . . . . 7 ((𝑅 Or 𝑆 ∧ (𝐵𝑆𝐴𝑆)) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴𝐴𝑅𝐵)))
64, 5mpan 690 . . . . . 6 ((𝐵𝑆𝐴𝑆) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴𝐴𝑅𝐵)))
76con2bid 358 . . . . 5 ((𝐵𝑆𝐴𝑆) → ((𝐵 = 𝐴𝐴𝑅𝐵) ↔ ¬ 𝐵𝑅𝐴))
8 breq1 5033 . . . . . . 7 (𝐵 = 𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶))
98biimpd 232 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶))
104, 1sotri 5961 . . . . . . 7 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
1110ex 416 . . . . . 6 (𝐴𝑅𝐵 → (𝐵𝑅𝐶𝐴𝑅𝐶))
129, 11jaoi 856 . . . . 5 ((𝐵 = 𝐴𝐴𝑅𝐵) → (𝐵𝑅𝐶𝐴𝑅𝐶))
137, 12syl6bir 257 . . . 4 ((𝐵𝑆𝐴𝑆) → (¬ 𝐵𝑅𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶)))
1413com3r 87 . . 3 (𝐵𝑅𝐶 → ((𝐵𝑆𝐴𝑆) → (¬ 𝐵𝑅𝐴𝐴𝑅𝐶)))
153, 14mpand 695 . 2 (𝐵𝑅𝐶 → (𝐴𝑆 → (¬ 𝐵𝑅𝐴𝐴𝑅𝐶)))
16153imp231 1114 1 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 846  w3a 1088   = wceq 1542  wcel 2114  wss 3843   class class class wbr 5030   Or wor 5441   × cxp 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-po 5442  df-so 5443  df-xp 5531
This theorem is referenced by:  supsrlem  10611
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